Problem 818

SET

The SET® card game is played with a pack of $81$ distinct cards. Each card has four features (Shape, Color, Number, Shading). Each feature has three different variants (e.g. Color can be red, purple, green).

A SET consists of three different cards such that each feature is either the same on each card or different on each card.

For a collection $C_{n}$ of $n$ cards, let $S (C_{n})$ denote the number of SETs in $C_{n}$ . Then define $F (n) = \sum_{C_{n}} S (C_{n})^{4}$ where $C_{n}$ ranges through all collections of $n$ cards (among the $81$ cards). You are given $F (3) = 1080$ and $F (6) = 159690960$ .

Find $F (12)$ .

神奇形色牌

神奇形色牌（SET®）由 $81$ 张不同的牌组成。每张牌有四个特征：形状、颜色、数字、阴影。每个特征有三种可能，例如颜色可以是红、黄、绿之一。

一个组合由三张牌构成，且对于任意特征，要么三张牌完全相同，要么三张牌完全不同。

考虑由 $n$ 张牌组成的特定牌组 $C_{n}$ ，记 $S (C_{n})$ 为这些牌可以组成的组合的数目。再记 $F (n) = \sum_{C_{n}} S (C_{n})^{4}$ ，其中 $C_{n}$ 遍历所有从 $81$ 张牌中取出 $n$ 张所能组成的牌组。已知 $F (3) = 1080$ ， $F (6) = 159690960$ 。

求 $F (12)$ 。